Quantum Mechanics is known for having a fundamental element of randomness to it. If you try to measure the position of a particle it is impossible to predict exactly where you will find it. That's because it was in several places all at once, a probability wave smeared through space. When we observe the particle, it "collapses" to a single location.
But why can't we just say that the particle was there all along? Maybe we couldn't know exactly where it was. But that doesn't mean it didn't have an exact location prior to measurement. What gives? Why do we need all this metaphysical junk about a particle existing in several places at once?
It turns out that there's a fairly good scientific reason for this. It was most famously demonstrated in what is known as Bell's theorem. Bell's theorem looks at the correlations between the spins of two entangled particles. I think it will help to cover some background first, so we understand the setup.
In classical physics, the spin of an object tells you how quickly it is rotating around its own axis. The Earth, for instance, spins around once per day. We can describe the spin as a vector, which has a magnitude and direction. The magnitude tells you how quickly it is spinning, and the direction tells you where the axis of rotation is pointing.
Spin in quantum mechanics is analogous, but there are a few wrinkles. We can measure the magnitude of the spin without much problem. Electrons always have the same magnitude of spin, no uncertainty about it. However, we cannot fully know the direction of the spin, because the uncertainty principle prevents us. We can only measure one component of the spin at a time. For example, if I measure the vertical component of an electron's spin, I will always find that it is "spin up" or "spin down". If I measure the horizontal component of the electron's spin, I will always find it "spin left" or "spin right". But an electron cannot simultaneously be spin up and spin right, no more than it can simultaneously have a definite momentum and location.
This is relatively easy to demonstrate with the Stern-Gerlach experiment. If you shoot a bunch of electrons through a magnetic field, their path will be deflected by an amount which is proportional to the vertical component of their spin. This experiment showed that electrons are always either spin up or spin down, and never in-between. A few modifications show that the vertical and horizontal components of spin cannot be simultaneously known.
In quantum mechanics, we describe particles as having quantum states. For example, we might say that an electron is in a state of 100% spin up, meaning that if we measured the vertical component of its spin, we would always find it to be spin up. We could also have an electron which is 50% spin up, and 50% spin down, meaning that we have a 50-50 chance of measuring either outcome.
When we have two particles, we do not describe them as each having a distinct quantum state. We instead talk about the quantum state of the whole system. For example, we could have a state where there is a 50% chance both are spin up, a 50% chance that both are spin down. If we measured the first electron's spin, there would be a 50% chance that it was spin up. Whatever we measure in the first particle is guaranteed to also be the result of the second particle. Unlike independent coin flips, the two particles' states depend on each other. This is called quantum entanglement.
This works even if the two entangled electrons are very far apart! In principle, we could send two entangled electrons to opposite sides of the galaxy. I could measure the first electron on this side of the galaxy while you measure the second electron on the opposite side of the galaxy. As soon as I find it to be spin up or spin down, I know what result you would get too. This is strange because if you were to send me a message from across the galaxy, it would take 100,000 years to reach me at the speed of light.
This "action at a distance" is known as the EPR Paradox, but in fact there is no paradox. Einstein's relativity requires that no information travels faster than light, because that would imply that you can send a message into the past. However, quantum entanglement does not allow you to send any information. The result of my measurement is random, and so is the result of your measurement, on the other side of the galaxy.
First, we need a source of entangled pairs of electrons (or photons, if we like). I'm not really sure what people use for an entangled electron source these days, but I'm fairly sure that it's nothing too exotic. Each of the electrons goes into a separate detector. If both detectors are measuring the vertical component of the spin, then they will always get the same results. Either both will be spin up, or both will be spin down. Therefore, we would say that the results are identical 100% of the time.
But what happens if one detector is measuring the vertical component, while the other is measuring the horizontal component? How much correlation is there with the two electrons?